Micky van de Ven emerged as the first defender to score twice in a English top-flight match for Spurs since Jan Vertonghen in March 2013.
Key points
The Spurs defender heads home two times in opening period
Everton's defender has equalizer ruled after VAR review
Senegalese midfielder scores final goal in 89th minute
The defeat is the Merseyside club's first at Hill Dickinson Stadium
The Tottenham defender scored on two occasions as Tottenham extended their winning streak in away games in the Premier League and handed the Toffees their first defeat at their new home.
Everton were unbeaten in their last five matches at their recently opened stadium at the dockside location, having departed their historic home behind after more than a century at the end of the previous campaign.
But Spurs captain Van de Ven struck on two occasions from set-pieces in the first half to put their strong form at jeopardy, and they were unable to mount a comeback before the midfielder - assisted by former Toffees attacker the Brazilian - nodded in a third to seal the victory.
Jake O'Brien believed he had levelled the tally a short while after the defender's first goal, glancing a powerful header past the Spurs keeper from an set-piece.
Beto, in the meantime, failed to convert a handful of opportunities, while the Italian goalkeeper made a series of fine stops to guarantee the London club went five away games unbeaten in the English top flight.
But the VAR stepped in, advising the match official to review the score on the pitchside monitor and the referee ruled that the winger and Iliman Ndiaye were in offside positions and had impeded with the Spurs keeper.
This victory elevates Thomas Frank's side into third place in the division, a handful of points off local foes and table-toppers the Gunners.
Everton stay in 14th position and sit six points ahead of the drop zone.
After a uninspiring midweek display in the European competition against the French side, where they mustered a only attempts on goal, Frank chose Randal Kolo Muani in attack instead of ex-Everton forward Richarlison.
The summer signing was unable to influence the game before being substituted for Richarlison after an hour on the field, but it emerged he wasn't needed in the final outcome.
Defender the Dutchman converted into an attacking force to be reckoned with.
Spurs' first goal came from a well-worked set-piece play, with the midfielder directing the ball back across the six-yard box for an unmarked the defender to head home.
And the captain - stepping up as captain in the Argentine's absence - added his second in added time as he rose above the England keeper to head home.
The Dutchman was just as impressive in defence, partnering effectively with Kevin Danso as the duo handled superbly with the numerous deliveries sent their way by the opposition.
Everton's new home had been a fortress, but it seems the freshness may have faded slightly and fans flocked to the exits after the midfielder's third for Spurs.
The end in the team's unbeaten home run will not be a major concern for David Moyes, but their attacking woes will be.
The forward and the other striker - the Everton's pair of recognised strikers - had scored only one league goal combined before Sunday's outing.
And they were unable to add to that poor record on a damp and dreary night on in Liverpool.
Beto's performance was summed up when he chose to overhead kick a golden opportunity from close range, even with having space and time to readjust for a more conventional shot that would have challenged the keeper.
The substitute, as well, failed to influence the game after coming on for Beto in the 65th minute.
It was not for a shortage of chance. The majority of the home side's offensive moves came through the winger and Ndiaye, who relentlessly whipped threatening crosses into the Tottenham box.
Everton: The team travel to Sunderland in the top flight on Monday, 3 November at 8 PM.
Tottenham: Spurs travel to Newcastle's ground on late October for their next round Carabao Cup tie against the Magpies.
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